Ternary Exclusive Or Francis Jeffry Pelletier, Department of Philosophy, Simon Fraser University, Burnaby B.C., Canada V5A 1S6. email: jeff[email protected] Andrew Hartline, Department of Philosophy, Simon Fraser University, Burnaby B.C., Canada V5A 1S6. email: [email protected] Abstract Ternary exclusive or is the (two valued) truth function that is true just in case exactly one of its three arguments is true. This is an interesting truth function, not definable in terms of the binary exclusive or alone, although the binary case is definable in terms of the ternary case. This article investigates the types of truth functions that can be defined by ternary exclusive or, and relates these findings to the seminal work of Emil Post. Keywords: truth functions, Emil Post, functional completeness, or. 1 Introduction Binary inclusive or is the (two-valued) truth function described in the table ϕ1 ϕ2 (ϕ1 ∨ ϕ2) T T T T F T F T T F F F That is, a sentence whose main connective is ∨ is true just in case at least one of its two arguments is true, and is false otherwise. Elementary logic textbooks point out that ∨ is associative, and so internal parentheses of complex formulas made up entirely of ∨’s can be dropped. It is also often pointed out that ∨ is commutative, so that all orders of stating the arguments are equivalent. Together these two facts are sometimes used to justify a “prenex” form of ∨ that is seen as having variable adicity: ∨(ϕ1, ϕ2,...,ϕn) which says “at least one of ϕ1, ϕ2,...,ϕn is true”. Binary exclusive or is the (two-valued) truth function described in the table ϕ1 ϕ2 (ϕ1 ⊕ ϕ2) T T F T F T F T T F F F That is, a sentence whose main connective is ⊕ is true just in case exactly one of its two arguments is true, and is false otherwise. In particular, it is false when both of its arguments are true. ⊕ is associative, so internal parentheses of complex formulas made up entirely of ⊕’s can be dropped. It is also true that ⊕ is commutative, so that all orders of stating the arguments are equivalent. Together these two facts can be used to justify a “prenex” form of ⊕ that can be seen as having variable adicity: L. J. of the IGPL, Vol. 0 No. 0, pp. 1–9 0000 1 © Oxford University Press 2 Ternary Exclusive Or ⊕(ϕ1, ϕ2,...,ϕn) However, this formula is not correctly understood as saying “exactly one of ϕ1, ϕ2,...,ϕn is true”, as can be seen from the simple case of n=3. ϕ1 ϕ2 ϕ3 ((ϕ1 ⊕ ϕ2) ⊕ ϕ3) T T T T T T F F T F T F T F F T F T T F F T F T F F T T F F F F Here we see that ⊕(ϕ1, ϕ2, ϕ3) is true not only when exactly one of ϕ1, ϕ2, ϕ3 is true but also when all three of them are true. So the argument form that was used in the inclusive or case to project “is true if at least one of the arguments is true” from the binary ∨ to the prenex variable adicity ∨ is not in general a valid form of argumentation: it makes use of some other features of the inclusive or that are not present with exclusive or. The reason that ∨ does intuitively allow for a variable-adicity form but ⊕ does not n is related to the fact that there is a “natural” definition of ∨ (ϕ1, ϕ2,...,ϕn), for n each n, but the same “natural” definition of ⊕ (ϕ1, ϕ2,...,ϕn) does not yield the meaning that it is true just in case exactly one of the arguments is true. For the former we have1 2 ∨ (ϕ1, ϕ2) = (ϕ1 ∨ ϕ2) n n−1 ∨ (ϕ1, ϕ2,...,ϕn) =df (∨ (ϕ1, ϕ2,...,ϕn−1) ∨ ϕn) And with this general definition of ∨n for all n, we might thereby justify the variable- adicity version, which we symbolize just as ∨, and say it means that “at least one of the following is true”. But as we have just seen, when this type of inductive clause is used for ⊕n, the resultant formula does not say “exactly one of the following is true”. In the case of n = 3 it asserts that either exactly one of the following is true n or else they all are true. More generally, ⊕ (ϕ1, ϕ2,...,ϕn) is true just in case an odd number of the arguments ϕ1, ϕ2,...,ϕn are true. This is called the property of being an odd counting function of adicity n, in the terminology of [4]. For n ≥ 2, all of the ⊕n are odd counting functions. It is shown in [4] that any composition of odd counting functions is itself an odd counting function.2 Since an n-ary connective (n ≥ 3) that is true just in case exactly one of its arguments is true would not be an odd counting function (for example if all three arguments were true then the function would be false), it follows that no such connective can be defined by ⊕.3 2 Meanings of Or It is standard in logic textbooks to distinguish inclusive from exclusive or – to dis- tinguish the binary connectives ∨ and ⊕. And there is a cottage industry in the philosophy of language and formal semantics literature on whether there is or isn’t an 1 1 Alternatively, we could start with ∨ (ϕ) = ϕ. 2 With a caveat concerning “dummy arguments”. For details see [4]. 3 n Nor by any of the particular ⊕ ’s alone, since they are all counting functions, nor by any combination of them. Ternary Exclusive Or 3 exclusive or in natural language. But another cleavage, more relevant to the present paper, is a distinction between meanings of the phrase ‘exclusive or’. Besides the issue of adicity of or, there is also the issue of whether we think of exclusive or as meaning ‘exactly one of the following’ or as meaning ‘being connected by ⊕’. If we think in the former way, we will be unhappy with iterations of ⊕, for the reasons we have just surveyed. Indeed, iterating ⊕ amounts to addition modulo 2, and while there are plenty of reasons to like this operation, it doesn’t express the ‘exactly one of the following’ sense of exclusive or. For the ‘exactly one of the following n items’ sense of exclusive or we will use the symbol ⊻n; and we will use the symbol ⊻ for the variable adicity version of this sense; we thereby distinguish ⊻ from both ⊕ and ∨. An interesting confusion arises because the two notions of exclusive or agree for 2 2 n = 2: ⊕ (ϕ1, ϕ2) and ⊻ (ϕ1, ϕ2) express the same truth table. One might want to say that the two notions of exclusive or, ⊕ and ⊻, are “extensionally equivalent” in the binary case, but are “intensionally distinct” because their extensions in other adicities are different. The logic textbooks concentrate primarily on the binary case, and so they do not distinguish these two connectives. In the case of natural language, it seems pretty clear that when people employ an exclusive or, as in “You can have either the steak or the chicken or the fish dinner”, they are using the ‘exactly one’ sense of or and not the ‘addition modulo 2’ sense. But there are also uses for iterations of ⊕ (that thereby define ⊕n). A natural use is in establishing the parity of an n-bit message, so that this information can be sent along with the message and the receiver can determine whether there has been some error in transmission by comparing the parity of the message with the extra sent bit. It also can find use in fast adders, where a simple ⊕n gate can determine the bit value of adding n-(binary)-numbers. (See [2, Chap. 5] for details). Our interest is in the formal properties of the ‘exactly one’ sense of exclusive or, since that topic has not been addressed by the logic textbooks (nor by the formal semantic descriptions of natural language). We will be calling this connective the “real” variable-adicity exclusive or, meaning thereby that it is the one that is relevant to formal accounts of natural language. We think that ⊕n might better be called “the odd counting function of adicity n”, and that iterations of ⊕ should be called “addition modulo 2” rather than “exclusive disjunction”. 3 Defining ⊻ Of course, with a functionally complete set of connectives like {∨, ¬}, {→, ⊥}, {↑} (nand), or {↓} (nor), we can define any connective. So, we could use such a set of connectives to define all the ⊻n, for any n. And if one allows variables in the syntax, one can even give a formula that expresses the general claim: ⊻n n (ϕ1, ϕ2,...,ϕn) =df (∨ (ϕ1, ϕ2,...,ϕn−1) ∧ ^ ¬(ϕi ∧ ϕj ) i<j≤n But that is not the method we want to pursue here. Intuitively, there should be some way to use some one exclusive or to define all the other exclusive ors without recourse to other connectives. That is, there should be some analogue to the method we em- ployed to define all the particular inclusive ∨ns, and thereby employed in accounting for the variable-adicity inclusive ∨. 4 Ternary Exclusive Or The key is to start with a ternary exclusive or rather than the usual binary ex- clusive or. We designate this connective as ⊻3; primitive formulas with this as main 3 connective take the form ⊻ (ϕ1, ϕ2, ϕ3). Using only this connective we can define ⊥ (the constant false) as follows: 3 ⊥ =df ⊻ (ϕ1, ϕ1, ϕ1) 3 (If ϕ1 is true, then there are three true arguments to ⊻ , and hence it is false; if ϕ1 is false, then there are zero true arguments to ⊻3, and again it is false.) The usual binary exclusive or, which we above simply called ⊕ when we used it as a normal infix operator (ϕ1 ⊕ ϕ2) and which we identified with the prefix operator of 2 2 adicity 2, ⊕ (ϕ1, ϕ2), would be called ⊻ (“exactly one of the two arguments is true”) in this new notation.